The EViews 2SLS output produces the F-statistic for the significance of the regressors in the 2SLS specification, not for the first-stage. Bearing in mind that I am not certain what Stata is doing, if they are producing an F-statistic based on the first stage, it would presumably be quite different.
I just noticed that Startz posted a reply to your original question. Coincidentally, I was going to provide you with a reference:
Startz, Richard, "Computation of Linear Hypothesis Tests for Two-Stage Least Squares," Economics Letters 11, 1983, pp. 129-131.