UK driving licence

Yes, the UK driving licence is still valid in Romania although who knows when the rules will change. Once your UK driving licence expires (or preferably before it does), you'll have to change it for a Romanian one if you're still living here. A lot of Brits have switched over to the Romanian one anyway, just to be on the safe side. At the moment, it's relatively easy to do and requires no re-taking of the test - basically just a straight exchange with a few additional documents - ID, translation of old DL, note from GP, etc.

To get Romanian citizenship you have to have lived here for eight years (less if married I think). Then you submit your file which includes quite a lot of paperwork and wait to be called for an interview (if accepted). At the interview, which is in Romanian obviously, you'll be tested on your language skills as well as questioned on the constitution, geography, history and culture of Romania.

However, I suspect your boss might be referring to residency, rather than citizenship, and wants you to get it so you have a 'CNP', which is basically like a national insurance number and is required for workers/tax purposes.

If you arrived in Romania before 31/12/2000 and picked up your registration certificate  ('certificat de inregistrare') under the old EU laws, then obtaining Romanian residency is pretty simple. If you didn't bother registering when you arrived (you're supposed to do it within three months of arrival) then you'll essentially be applying as a non-EU citizen, which is more complicated. I'm assuming here that you're British as you asked about the UK driving licence but if you're a passport holder of another EU country, then residency is very easy.