**Additional Info**


There are several variants of the $\chi^2$ test. But what is common between them is that they refer to **$\chi^2$ distribution**. Variances, which are always positive cannot be a normally distributed. These follow $\chi^2$ distribution. The F-test for variances uses the ratio $\chi^2$ statistic of the two random variables denoting variances. Even in the Pearson $\chi^2$ test, the test statistic is a sum of squares which makes it always positive. 
In fact this $\chi^2$ distribution is also used in t-test. As . Kennard said, one of the assumptions of t-test is that the population variance is unknown but assumed to be equal. Since population variance is unknown it has to be estimated from the sample. Like the case with all estimates you dont have a fixed value but a range of acceptable values falling in some confidence intervals. T-distribution is basically an average of several normal distributions with variance values falling in the allowed confidence interval of a $\chi^2$ distribution. 

Bottomline: a statistic tested by a $\chi^2$ test is a square- something that can never possibly have a negative value.