Hi,
AFAIK, SWT is a set of Java UI controls and Eclipse is a shell to integrate different tools.
TestComplete definitely supports SWT controls.
As for the Eclipse - do you need to automate Eclipse itself or wondering about automation of the application compiled from within Eclipse IDE?
If the former, then I am pretty sure this is possible though never tried it.
If the latter, then it depends on the application itself and used compiler (and Java applications can be automated using TestComplete) but not from the IDE used to build the tested application.