For my position a bachelor of science in a basic science was required. I had very little laboratory experience when applying, which I'm confident was a negative. What I did do on my resume which I think worked very well was a "Relevant Laboratory Experience" section where I listed some of the procedures (PCR, gel electrophoresis, sample lysis...) we had done in our general science labs. If you have relevant experience, great--make sure you list that prominently. Otherwise I suggest doing something like I did.
I also recommend that you apply broadly. In the end I think I applied to over 100 positions. Of those, I probably only heard from 20, period. Of those 20 I was invited for 4 interviews. Of the 2 of those interviews I took, one offered me the job (I would've taken the other two if I were still looking). I also applied to pretty major medical universities, though (Northwestern, UChicago, Rush, Loyola). I recommend you go on their websites where they list research technologist/technician/assistant positions and apply to every one you're remotely qualified for.
You're also a little early. I might hold off until the beginning of April. Finally, being able to relocate from home and apply nationally would help too. Good luck. My job has been great experience.