It seems to me that they are basically calculate the same thing. Since we already have t-test, why do we need a squared version (wald test)? Does wald test have its own advantage? For example in Cox model, why can't we just use t test?
There are similarities between a Wald test and a t-test, as this page describes.* They both are used to determine whether a coefficient value is significantly different from 0 (or from any value specified in a null hypothesis). They typically arise in different contexts that show why we need both.
Consider the situation for testing whether a coefficient is significantly different from 0 in two scenarios: one for the mean value of a set of numbers examined with a t-test, and another for a coefficient in a Cox regression examined with a Wald test. In both cases you use the ratio of the observed difference from 0 to a measure of the error in that estimate. The error term, however, is determined in different ways.
For testing whether a mean value of a set of numbers is different from 0, the t-test takes into account the fact that you are estimating both the mean value and the variance from a particular number of observations. It assumes that the underlying population is normally distributed and uses the properties of normal distributions to calculate the error estimate. This gives an exact answer, under that assumption, for the probability that the observed difference of the mean value from 0 could have arisen by chance if the true population mean were 0.
The error estimate for coefficients determined by maximizing likelihoods (partial likelihoods with Cox regressions) are determined in a different way. This answer describes how the matrix of second derivatives of the log-likelihood, calculated at the maximum likelihood, can be transformed into an estimate of the variance-covariance matrix for all coefficients in a multiple regression setting.
For the case of a single coefficient that matrix is a single value, the variance of the coefficient estimate. The Wald test for a single coefficient assumes that the coefficient estimate is normally distributed. That's different from the assumptions for the t-test; in this case you aren't sampling a specified number of times from a normal distribution but rather using your estimate of the variance directly. Note that with small sample sizes that assumption of normally distributed coefficient estimates might not hold.
This page shows how the t-test and the Wald test become equivalent as sample size increases.
*The form of the Wald statistic that you have in mind, for testing against a chi-square distribution, is the square of that presented on the linked page.