The Gregory-Hansen and the Enders-Siklos tests are both residual based test. The formal theory of Philips and Ouliaris (1990) showed that for the linear cointegration test, critical values depend on the number of regressors. For the two non-linear versions you are interested in:
Gregory-Hansen: their formal derivations show that the statistic
depends also on the number of regressors. They provide critical
values, but only up to 4 regressors
Enders-Siklos: they do not provide a formal theory, and the critical values are derived from a DGP with 1 regressor.
So in both cases, you will have difficulties using these tests with 10 regressors. What you could do is to replicate the authors Monte Carlo simulations, and derive yourself the critical values for 10 regressors. Or maybe you will find more recent work allowing for more regressors (look maybe at Kejriwal and Perron for structural breaks?).
As an aside, note that residual-based tests apply when you have multiple variables but one single cointegrating relationship. Is this what you find in your data?