Why can Thomsen's parameter $\epsilon$ be negative in VTI media?
Negative values are physical and are expected. 

Indeed, Thomsen's 1986 paper has several negative values for $\delta$, $\epsilon$, and $\gamma$ — look at the data in Table 1. He also discusses negative values, at least for $\delta$ (e.g. on page 1961) . You can quickly get a sense of how common negative values are from his Figure 4:



As for physical meaning, it depends on the parameter. I recently wrote this about the physical meaning of the Thomsen parameters (edited for brevity): 


  
  $\delta$ or delta — the short offset effect — captures the
  relationship between the velocity required to flatten gathers (the NMO
  velocity) and the zero-offset average velocity as recorded by checkshots.
  $\epsilon$ or epsilon — the long offset effect — is,
  according to Thomsen himself:  "the fractional difference between
  vertical and horizontal P velocities; i.e., it is the parameter
  usually referred to as 'the' anisotropy of a rock".
  $\gamma$ or gamma — the shear wave effect — relates, as rock physics
  meister Colin Sayers put it on Twitter, a horizontal shear wave with
  horizontal polarization to a vertical shear wave.
  


Usually these parameters are, at a gross scale anyway, positive, because usually velocity is faster along bedding (roughly horizontally) than across it (roughly vertically). But it's easy to imagine scenarios where the relationship is reversed, especially if we're talking about small scales — as in log measurements.

Intuitively, we can draw these conclusions about negative parameters:


A rock with $\delta &lt; 0$ has the zero-offset reflections 'seeing' faster velocities than the long-offset reflections.
A rock with $\epsilon &lt; 0$ has a faster vertical velocity than horizontal velocity.
A rock with $\gamma &lt; 0$ has a faster vertical shear velocity than horizontal shear velocity.


Reference

Thomsen, L (1986). Weak elastic anisotropy. Geophysics 51 (10), 1954–1966. DOI 10.1190/1.1442051.