Not currently.
Hosted Business Messaging is currently built on old messaging infrastructure that does not support file sizes of modern videos and even some pictures.
In order to reduce the file size low enough to be sent successfully, the video would be downsampled so much it is border line unwatchable. Think of watching videos on the old Blackberry phones.
There is new technology coming out called RCS (Rich Communication Services), that will bring many features from systems like iMessage to business texting.
We are currently exploring all of the benefits with our carrier in a closed beta and this new technology will enable better videos.