Before analyzing the systems of Amyraldism and neo-Amyraldism, it would be interesting to note Piper's rationale for holding such a view in the first place.
In terms of rationale, Piper is on record in his article for saying that the reason why he postulated that God does indeed desires the salvation of all is primarily because of dissatisfication with the exegesis of typical Arminian prooftexts offered by Calvinists. This type of thinking is totally unfounded, especially since he offered no exegetical proof at all why the traditional Calvinist understanding of these texts are wrong, and gives perhaps a hint that Piper is reading into Scripture here instead of reading out of Scripture, which I would prove by dismantling his system.
The commonality between the Amyraldian and the neo-Amyraldian system is the belief that God has two wills with different saving desires and intentions. On the surface, this makes God a schizopreniac as these two wills operate against each other in one person. Piper has correctly pointed out, however, that the reality is that there are at least two-fold wills in God which are stated in Scripture,, and thus the charge does not stick on the surface. However, this charge does stick at a deeper level against both neo-Amyraldism and Amyraldism, as unlike the other examples given of the two wills of God, the two wills of God mention here are both at the same level of God's active willing with regards to soteriology; one part of God's will wills to save all while the other part doesn't will to save all but only the elect. Whereas in the example of Christ's death, God did not actually 'will' that the people obey the Law and thus not put Him to death (He decreed in the Law, but it was a command rather than a will), and thus the paradox can be resolved by stating that it was God's command according to the Law that Jesus would not be crucified, but that it was God's will to put Him to death. The Amyraldian could not resolve the paradox they have created in any logical manner, and therefore their position face a logical impossibility.
Besides being illogical, let us look into the implications of such a view on our view of God and the logical heresies that follow.
The first implication of the 'two wills of God' theory is that God would be forever unhappy and miserable since one of His will is destined to be forever frustrated. As Dr. C. Matthew McMahon says,
This would make God sin. He would sin in that He would violate His own mind and omniscience. He would go against that which He knows is true. He would desire the salvation of men which He will never regenerate. This would make God frustrated. He would be the ever-blessed, ever-miserable God.
Another implication would be that this would create a conflict between the will of God the Father and the will of God the Son. This would be a serious problem for classic Amyraldianism since in that system, the Atonement was intended by God the Father for all Man. However, Christ's atonement according to His expressed will is only for His sheep and not for the whole world (Mt. 25:33; Jn. 10:15). Amyraldism is therefore a logical absurbdity and contradicts Scripture.
With regards to neo-Amyraldism, the latter problem is of course, avoided. However, it still faces the logical coherence problem and the problem created by the implication of such a view on the nature of God Himself, which is enough to invalidate the entire theory already as abberational.
Now, with regards to Piper's take on the typical Arminian prooftexts, the only thing that seems likely to support his interpretation is that on the passage of Ezekiel 18:23, 32. Although the passage and verses are directed towards Israel which is symbolic of the people of God of all ages, yet the verses cannot be limited to just saying that God does not delight in the death of the wicked within the communion of believers. Since the death of the wicked within the visible communion of believers in that state would mean that these people are not saved in the first place (1 Jn. 2:19; Heb. 12:14), this means that God does not delight in the death of at least certain reprobates, and the word 'anyone' in Ez. 18:32 makes this action of 'not delighting in the death of the wicked' to be applied to an indefinite number. So, I can agree with Piper's exegesis on these two verses. However, to infer from the fact that God does not delight in the death of the wicked to the theory that God desires to save all is an unfounded leap in logic which I totally reject.
From all this, we have refuted both Amyraldism and neo-Amyraldism according to logic and Scripture. We will look now more into detail into the theories derived from it; 'Common Salvific Grace' and the 'Free Offer of the Gospel'.
[to be continued]
 Are there two wills in God? (http://www.desiringgod.org/ResourceLibrary/Articles/ByDate/1995/1580_Are_There_Two_Wills_in_God/)
 I wouldn't use the term "will"' to describe things like "God's moral will", as I think it is not proper terminology, but I do agree and subscribe wth the meaning and intent of the terms used.
 Hypothetical Universalism (http://www.apuritansmind.com/PuritanWorship/Amyraut%20Universalism.htm). Although this article is a good resource, it is my opinion after reading though it that there are a few errors in it. Amyraldianism does NOT believe in all man having a equal chance to 'become Sons of the living God', at least not in the manner understood by Semi-Pelgianism or Arminianism, contrary to what McMahon thinks it believes. Another thing to take note is that McMahon tend towards hyper-Calvinistic thought patterns in his disregard of the sufficiency of the Atonement for all who would believe (Particularism), which is deduced from Scripture passages such as Jn. 3:16 whereby the atoning sacrifice of Christ is said to be effective for an indefinite number 'all', 'the world'.
 I DO agree that God ultimately delights in the destruction of the reprobates due to their sins, and Piper does too, and such delight also occurs on this earth (1 Sam. 2:22-25). What I believe regarding this issue is that God does not delight in the death of the wicked when viewing them as humans, but God delights in the destruction of the wicked when viewing them as depraved rebel sinners.