Is Rectilinear Steiner Tree still NP-complete when points have integral coordinates?
Yes.

I suspect that you have not read the paper The Rectilinear
Steiner Tree Problem is NP-complete, by M. R. Garey and D. S. Johnson, SIAM Journal on Applied Mathematics, Vol. 32 (1977), pp. 826-834. Well, if you have, I am sorry. Well, if you have, please indicate that critical information in your question. In fact, whenever your question is based on one big theorem/one paper, it is always a good idea to inform readers here whether you have no access to or have read or are familiar with a proof or proofs of the theorem/that paper. People, or at least I, will then be able to exchange ideas with you more efficiently in a way that suit you better.

I just browsed through the proof by Garey and Johnson. It is clear that, in fact, they build their proof on the rectilinear stree problem with integral coordinates within a square of side length $n^{O(1)}$, where $n$ denotes the number of points.