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I am looking at logistic regression output that shows Z test values. When can I use these to estimate the significance of coefficients, and when do I need to use a Wald chi-squared test?

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They are almost certainly the same thing. The "Z test values" in your logistic regression output are probably Wald statistics, and the Wald chi-square statistic is probably just $Z^2$. This means that the Wald chi-squared test is the same as a two-sided test using the corresponding $Z$-value.

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    $\begingroup$ So just to be clear - if I need the Wald's Chi-Square value, all I need to do is take a square of the Z statistic from the output? $\endgroup$
    – Indrajit
    Commented Feb 5, 2018 at 8:15

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