Should I recommend a student with a bad ideology?
As stated in the other answers, no-one can coerce you to provide a recommendation, you are free to not recommend a student for any reason you choose, or no reason at all - if you are not comfortable recommending this student, then don't.

However, from your description, I would suggest that it's not the ideology at issue, but the student's interpersonal skills. I guarantee you that I have opinions that you would find absolutely abhorrent, and you have opinions that I would likewise. The point is that in a professional context we usually have the good sense to keep these things to ourselves. 

It's not clear what your opinion of the student, and their suitability for the role, actually is. If you think that the way he expresses his ideology interferes with his ability do the job well, or his ideological standing is in some way relevant to the job at hand, then there is no conflict here. (For example, a raving misogynist would be completely inappropriate in a supervisory position.)

If you think that he nevertheless really would be an asset to the company, then you can either (i) informally suggest to your contact: "Here is a brilliant student who will produce technically excellent work, but be careful; he has a bit of a problem with appropriate boundaries at work." and/or (ii) write a letter that expresses his technical skills and omits any mention of personality or social skills. The omission will be  loud and clear. In the end, it is up to the company to decide if the tradeoff is worth it.