I have written several times on the biblical case for particular atonement, so I won't attempt to do so again here. Use the tag at the bottom to go to those posts. Rather, here I will deal specifically with Amyraut's attempt to remove the doctrine from the Calvinist system.
Amyraut's view boils down to this: 1) the Father has foreordained a certain number of individuals to salvation; 2) the Son died to atone for every sin of every member of the human race, without distinction; and 3) the Holy Spirit applies that atonement only to the elect. I think that the obvious reaction to this combination of precepts is that the Persons of the Trinity are put at cross purposes. However, the orthodox view of the Trinity forbids this. And Scripture concurs.
Jesus testified that He, the Second Person of the Trinity, does only the will of the Father, the First Person. See Luke 22:42 and John 5:19-24. And He further testified that the Holy Spirit, the Third Person, acts only in accord with the word of the Son (John 16:7-15). To have any contrary acts or intent within the Trinity is thus impossible. I consider this flaw to be fatal to any system of Calvinism which tries to incorporate the Arminian doctrine of universal redemption (except for the one who holds to universal salvation, which is a separate issue altogether.)