Translation:My boyfriend was employed at a school.
Not necessarily, depending on context and which part of the sentence you want to emphasise. However, honestly, I'm very sure that most people would simply use 'at school' (at least in English, regardless of country), since adding an article (a, an, the) is usually only used to specify something or to make something stand out (in Japanese, the particle "ga" can sometimes be used, similarly). Additionally, even with the article added, you still need context to confirm whether or not he's attending the school and working on something related (such as, if he's a teacher, cafeteria worker, etc.), or otherwise.
If, in this example, you wanted to make clear that the boyfriend was working on unrelated material while physically being at/inside of the school, either an adjective/adverb and/or a direct object is the best way to go about it (e.g.: My boyfriend was working on his paperwork while at school.).
Technically, 仕事をする(shigoto wo suru), 働く(hataraku), and 勤める(tsutomeru) are all ways to discuss work/jobs. However, there are a few subtle differences, just as in English:
仕事をする(shigoto wo suru) - used to describe your current and possibly temporary work status (not something that is necessarily a career) and is good to use when giving a general description of what you do or for naming a set position in your current job (e.g.: 'Well, right now, I'm currently working for a company in the accounting department.')
働く(hataraku) - used to simply state that you are performing work/a job, regardless of what you want to do for a living or may do for work in the future (e.g.: I have to go to work at 8:00, today.')
勤める(tsutomeru) - used to describe long-standing jobs and careers (e.g.: 'I've been in engineering for ten years.')
Basically, 仕事をする(shigoto wo suru) is most useful for any actual description of what you do or to generally state that you are working; 働く(hataraku) is best for discussing stable yet not necessarily specific labour, regardless of whether or not it's your "job"; 勤める(tsutomeru) should be used when talking about serious and/or permanent jobs/duties/careers; .