Can All-on-4 upper work with a crossbite? I do have some gingivitis.

If your cross bite is minimal, an all on 4 could possibly help. These questions can only be answered by completing a full head and neck examination and evaluation of your bite followed by X-rays to see if you have enough bone remaining

You should have a comprehensive evaluation which includes a periodontal exam. You referenced the top arch but as a surgeon I would like to understand the condition of your lower arch as well. It is critical important to know whether your upper All-on-4 would be biting down against natural teeth or another type of prosthetic. One of the main benefits of the All on 4 procedure, especially when done in both arches, is that the pattern of the bite can be corrected and possibly your cross-bite may be eliminated. Of course, this is all based on your question. A thorough exam and CT scan should take place followed by a treatment conference to explore and identify your existing conditions and go over risks associated with doing the procedure as well as not doing the procedure and a detailed appointment plan done. 

Hi:
Unfortunately without photo and X-ray it is difficult to give you accurate advice.  Please post photo and X-ray of the affected area.
Best,
Dr. Maddahi