In general, whether you tell your employer or not is not really a legal matter. A lawyer can only point you to your rights:
Is the background check pursuant to the FCRA (Fair Credit Reporting Act)? If so, you have a number of rights, such as that they need to give you a copy of the report, that they need to discuss it with you, et cetera.
Are you a minority or a member of a protected class? That makes a difference, too, because termination may violate federal law about discrimination.
Finally, what was your sentence? You may be able to expunge or seal the conviction as well. Speak with an attorney about doing so.
My two cents is that disorderly conduct is a very minor offense, and no reasonable employer would fire a good employee based on that.
This answer is intended as informational only, and does not constitute legal advice or form an attorney-client relationship between us.
Are you sire you received a conviction? What was your exact sentence? Did your employer ask if you had been convicted of a misdemeanor? If you received supervision then you were not convicted. If your employer has not asked, there is no need to volunteer information. Too few facts are presented here to answer you completely even though you provided facts about the incident. It appears you were not represented at the time. That was a mistake.
Consult and retain an experienced attorney ASAP.
Please do NOT use this answer/response to say or do anything regarding your situation. This answer/response is based on the information provided in the question asked and requires a much more complete context than is available in this public forum. BEFORE you say or do anything consult with an experienced Federal and/or state criminal defense attorney in your jurisdiction who will listen to you and your concerns.