In December, Javid suggested that public officials should swear an oath to protect British values.
But Christian Concern have said that the value of "equality" has been used to oppress Christians who oppose gay marriage.
Speaking on Premier News Hour, Christian Concern's Tim Dieppe said that the government has previously misinterpreted the idea of equality.
"The problem is how you define British values," Dieppe said, "Equality, in particular, has been used to oppress Christians in various situations, such as the Ashers case."
Ashers Bakery is currently trying to appeal against a ruling it discriminated against a gay activist by refusing to make his a cake with featured a slogan supporting gay marriage.
The lobby group fears that, if the government introduces the oath, it could bar Christians from being school governors, civil servants and MPs.
"If you're made to swear an oath to equality and then you say you don't believe in gay marriage or that children will be better off with a mother and a father where possible, you could say that doesn't go with the British value of equality and would essentially bar Christians from taking office," Dieppe said.
Christian Concern is calling on Christians to write to their MPs and voice their opposition to the oath.
Dieppe told Premier: "Society and the government are using this oath in an oppressive and discriminatory way and that's why Christians need to stand up and say that this [oath] is not a good idea."