Can employers see whether or not I paid taxes certain years in a
background check?
The answer is, it depends.
- Average employer, probably not. Their background check will not be that thorough.
- One that requires a more extensive background check, it is more likely that they will. If you are applying for a position that requires a financial check they could ask for tax forms.
- One that requires a government clearance, they will probably discover it.
Keep in mind that if you owe taxes, and you can't pay them, then the IRS or the state taxing authority, could garnish your wages. That would be a big clue that there is an issue.
If they would be concerned about non-payment: you are applying to be an accountant, then lying on the forms could get you fired.
Note most government tax agencies look for missing years. It could be a matter of time before they ask about the missing forms.
Edit:
From a "getting hired" perspective, would I be alright putting that I
worked on my resume even though I didn't file taxes?
Your resume lists your experience related to work, it is not a comment on your financial condition. Keeping in mind that your lack of filing and paying taxes may hurt your chances for some jobs, it is not relevant for other jobs.
The sooner you file the required tax paperwork, the sooner the issue can be resolved. And the sooner interest and penalties will stop accumulating. You need to get expert help.