They did not have a "BC" equivalent as far as I know however the time/calendar was divided in three eras. The "common era" (ère vulgaire) was prior to 1789 or 1792. Pretty much the Gregorian calendar and everything that happened before. 1789 marked the beginning of the "era of Liberty" or the "era of the French".

The era of Liberty was established in January 1792 (in the Third year of Liberty) by the Legislative Assembly. But the era of Liberty officially began after the Bastille fell (14 July 1789). The idea of creating a new calendar was not new and some people amongst the Revolutionaries worked on different ways to create a calendar. The proclamation of the Republic allowed for a complete "rework" of the calendar with the eradication of Christian connotations. The Republican calendar, although it starts on 22 September 1792 date of the birth of the French Republic, was established the same day. It was incomplete though so it became effective in October 1793 (Second Year).

Before 1789 it was the common era. Then until the Republic was proclaimed it was the era of Liberty (or/and Equality). However it not rare to see some official records mixing up dates between all these calendars like this one for example : "Today 9 Pluviôse 1794 year two of the One and Unbreakable French Republic".

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The 5 October 1793 decree is a good example of the use of these terms :

First Article

The era of the French starts with the foundation of the Republic on 22 September 1792 of the common era [...]

Article II

The common era is abolished in civil law.

I hope this answers your question.

EDIT : the term "common era" I use is the translation "ère vulgaire".