I will not deny that it is confusing but "Gender Critical" is the exact opposite of the Right Wing Christian "anti-gender" stance, which might be more accurately described as "anti-feminist".
Right Wing Christians are against what they call "Gender Ideology." This terminology is most common in Eastern Europe.
By contrast, "Gender Critical" means being critical of "Gender Identity Ideology".
Anti-women activists on both the Left and the Right sometimes deliberately conflate the "Anti-Gender Movement" with "Gender (Identity) Critical" views - but for very different reasons. On the Left, to smear "Gender (Identity) Critical" views as being aligned with the Right Wing Christian "anti-gender movement"; on the Right, to attempt forced-teaming between Right Wing Christians and Radical Feminists.
The anti-women trans activists who have a stranglehold on Wikipedia have, for political reasons, subsumed Gender (Identity) Critical views as an aspect of the "Anti-Gender Movement". They further reveal their political motivation by referring to people who hold Gender Critical views as "TERFs".
The reason why "Gender Critical" became popular as shorthand for "Gender Identity Critical" is that the vast majority of people who hold Gender (Identity) Critical views are so far removed from the "Religious Right" that they were completely unaware of the term "Anti-Gender", ie. as in "Anti-Gender Movement".
This confusion with the Anti-Gender Movement is why some people dislike the term "Gender Critical" and every now and again there is a thread on Mumsnet calling for a better description.
In 2019 a guest author on Fair Play for Women suggested "Critical Realism":
https://fairplayforwomen.com/critical_realism/
However, Maya Forstater's Employment Tribunal Appeal has established that Gender Critical beliefs are protected under the Equality Act 2010 Protected Characteristic of "Religion and Belief". The ruling also defines Gender Critical beliefs. These relate to Gender Identity.
So, in the UK at least, there is clear separation between the "Anti-Gender Movement" of the Religious Right and "Gender Critical" as shorthand for "Gender Identity Critical". The UK definition established in law does not make any reference to Religion in general, nor to Christianity specifically nor to any political persuasion.
So, No. Right-wing Christians do not "tend to be on the Gender Critical side". The Church of England aka "The Tory Party at Prayer" has bought into and preaches "Gender Identity Ideology" aka "Transgenderism". In the UK, Right-wing Christians tend to be on the side of magical gender essence and "gendered souls".
Gender Affirming Transgenderism, rather than "Gender Critical" is, naturally, "right-wing Christian", as well as "fundamentalist Islamic", as it is regressive, anti-science, anti-woman and anti-feminist.