Both clubs originally have ties to Methodism. You can find evidence on both sides for either being 'traditionally' one or the other but the evidence is always tenuous at best.
There is no Catholic or Protestant split these days between Everton/Liverpool, however. Liverpool has a huge Irish Catholic community and both clubs enjoy support from both denominations. Usually Everton and Liverpool fans would have been friendlier with Celtic than with Rangers though in Everton's case this has ceased to even be a talking point anymore. Liverpool probably have closer ties with Celtic these days owing to them having far more fans than in Ireland (clearly evidenced through their singing of YNWA and the Fields (of Athenry/Anfield Road)).
The best argument against any latent sectarian split between Everton and Liverpool is the far clearer and more famous intra-familial split. It's kind of impossible for there to be any sectarian split if support is split within families of the same religion, isn't it?