I don't know if OP would resign from one job without having another lined up, but I wouldn't.
That said, to accept the counteroffer after being told he wasn't eligible means breaking your acceptance of the other job. The other firm may have already let other applicants know that they weren't chosen. On top of that, the original company may start looking to lay you off when it's convenient for *them*. Plus, they've already shown themselves to be dishonest, because he wasn't eligible for a raise but suddenly was when their back was up against the wall.
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That said, to accept the counteroffer after being told he wasn't eligible means breaking your acceptance of the other job. The other firm may have already let other applicants know that they weren't chosen. On top of that, the original company may start looking to lay you off when it's convenient for *them*. Plus, they've already shown themselves to be dishonest, because he wasn't eligible for a raise but suddenly was when their back was up against the wall.