What does the molecular orbital scheme of beryllium chloride and hydride look like?
I calculated molecular orbitals for $\ce{BeCl2}$ at MP2/jun-cc-pVDZ level of theory; given below are the doubly occupied MOs at this level of theory along with their symmetry designations and energy in a.u. For some reason my Avogadro kept crashing when I tried to select an iso value I thought appropriate for the last two MOs; anyway these are completely localised $\ce{Be}$ atoms. If we look at the 8 valence orbitals, there is indeed delocalised $\pi$ bonding as you suggest, and do bear a striking resemblance to MOs for $\ce{CO2}$. 

For $\ce{BeH2}$ I again ran into trouble generating nice visuals using the cube file in Avogadro. I will update this post with visuals as soon as possible (It would be great if someone can help out).

I got the following doubly occupied MOs at MP2/jun-cc-pVDZ level of theory:      -4.644660 ($\ce{a_g}$) ;     -0.504366  ($\ce{a_g}$) ;  -0.469857  ($\ce{b_{3u}}$). However, it is evident from the symmetry designations that $\pi$-type orbitals aren't involved in bonding. 

P.S: If you are interested, I can provide the input, output, and cube files on request.