What does it mean if eye diagram voltage (0 and 1 level) goes negative?
To answer the question in your title,


  What does it mean if eye diagram voltage (0 and 1 level) goes negative?


You are looking at a system with differential signalling.

A differential signal has a negative voltage if the complementary node is at a higher potential than the "true" node. Said another way, for a differential signal \$V\$ composed of two signals \$V_p\$ and \$V_n\$, the differential signal is negative if \$V_n &gt; V_p\$.


  I get negative eye density while checking the eye diagram


I don't see any sign of negative eye density in your diagrams.

You have positive eye density at negative voltages. That's not the same as negative eye density.

Negative eye density would mean that a negative number of traces passed through a point in the diagram, which is simply not possible. 

I can think of cases where a certain analysis technique would produce a graph showing negative eye density due to noise and the limitations of the analysis technique, but that doesn't apply to your situation.