Checkstyle cannot do this out of the box. So if using a third-party library is no option for you, you will have to write a custom check. If it's any comfort: This would be quite a simple case of custom check.
The best you can do out of the box is to configure a DescendantToken check like this (works in Eclipse-CS; DescendantToken checks are complex low-level checks which require understanding of the AST to configure, please see the linked documentation for more information):
<module name="DescendantToken">
<property name="tokens" value="LITERAL_IF"/>
<property name="limitedTokens" value="LNOT"/>
<property name="minimumDepth" value="2"/>
<property name="maximumDepth" value="2"/>
<property name="maximumNumber" value="0"/>
<property name="maximumMessage" value="Consider changing this to a
positive condition for better readability."/>
</module>
This approach is somewhat crude, as it will not detect if there is an else block, and it also flags method calls such as if (!map.isEmpty()). But it covers your if (!employeeNotExist) case. It will also flag conditions like !(a || b), but those can be rewritten to !a && !b.
So, all in all, it's far from perfect, but this is the closest you're going to get with out-of-the-box Checkstyle.