Is there any female equivalent to ‘Esq’ or ‘Esquire’?

The short answer is no.

The long answer:

First, from Wikipedia, Esquire

British men invited to Buckingham Palace receive their invitations in an envelope with the suffix Esq. after their names, while men of foreign nationalities instead have the prefix Mr (women are addressed as Miss, Ms, or Mrs).[21]............

....In the U.S., the title Esquire is commonly encountered among members of the legal profession.[7] The term is used for both male and female lawyers.[39].........

....In the United Kingdom, Esquire historically was a title of respect accorded to men of higher social rank, above the rank of gentleman and below the rank of knight.

(These quotes are not in the same order as they appear in Wikipedia. Although it is not clear from the first quote, "men" refers to untitled men.)

Dame is the female equivalent of knight, Debretts, Dame: Dame Judi Dench; Sir Laurence Olivier.

To back up Wikipedia for the UK, I looked at Debretts. From Debretts, the section Forms of Address, Untitled Men:

Gentlemen

It is for the writer to decide whether to use one of the following three styles: John Brown, Esq, Mr John Brown, or simply John Brown.

Debretts has a section on Forms of Address, Untitled Women, which is too long to quote. They discuss only Miss, Mrs and Ms.

To back up Wikipedia on the US, see FindLaw, a short article on the difference between JD and Esq:

However, when choosing a lawyer, don't just rely on the "Esq." or the word "Attorney" after her name and assume she is licensed to practice.

Despite the perhaps PC way of indicating that Esq applies to female lawyers, I think it is beyond dispute that this is so, in the U.S., at least.