You may come to HK for 3 months purely as a visitor. You may not reside here, and you may not, in principle, come here for the purposes of looking for a job. The immigration rules are quite clear on this.
It is really quite hard to get an Employment Visa here unless you have specific skills or experience that are not available locally. The two most common ways for foreigners to move here are to come on an internal company transfer or to be hired as English teachers.
Obviously a few people do just roll up and somehow find a company that will sponsor them in a job for which Immigration will grant an Employment Visa, but my guess from postings here is that the success rate for people attempting that is less than 20%. The chances of both of you achieving it (which is, presumably, a condition of you staying here) would therefore be less than 5%.
One other option would be to apply for the Quality Migrant Admission Scheme which is allegedly getting easier to be accepted for, although we haven't heard of any first hand examples on this board as far as I know. However, since I surmise that you are running away from something (military service for your boyfriend?) rather than having any specific reason to come to Hong Kong you may find that difficult as well.