Yeh theyre pretty crucial. Its the only thing AdComs get to reassure them that you are the wonderful person you have made out in your essays.
Basically they want the recommendations to align to everything you have said in your essays - but they also want them to shower you in praise - state that you're the best employee theyve ever had and you're destined for greatness.
I'd definitely say identify the potential people you think could give you decent recs - ask them for a coffee or catch up and lay out what it is you're wanting to do and why etc. Explain you're trying to identify who could rec and thought they might be suitable - then have a list of th equalities the school is looking for and ask them if they are comfortable saying they think you would be an ideal candidate given those qualities (and that they can provide examples of this in their time spent working with you.)
If they agree then I'd advise you prepare them a recommender pack - basically the questions they will be asked and prompts as to the projects/work you have done with that person that shows those qualities the schools seeks to help them identify and be able to remember what exaclty you have done (especially if those recs are from a year or two back) - this will help them writing your rec and ensure that the things they speak about are consistent to your wider story. (prompts should be prompts - limited bullet points to jog their memory rather than detailed points where they might end up copy pasting!)
Whilst I understand why your boss may feel like that - that kind of attitude will result in him never building great teams. I was in a similar boat in that I was particularly keen not to use my current boss - she wouldn't take it the right way, and I would've kissed goodbye to all the interesting projects and promotion over the last 6 months.
Very fortunately I had a senior member of the team who has plenty of work experience etc who was very approachable and understood my aspirations - I therefore approached him about writing the reference which he did and given we worked closely on a number of projects he was able to give some very detailed obversations. If there is someone like this that you can use then that would be completely acceptable. BS's are aware that people might want to avoid using current bosses but giving them a more senior person who has first hand experience of your work will satisfy them. They will also be likely to give you a glowing reference without any fear of losing one of their 'employees'.
For your other rec - you kinda want to use someone else who has a sustained level of experience working with you so a year plus etc. It may be that you need to find two people from your current job like the above but without a more detailed work history it would be hard to make suggestions