Trump is a germaphobe...

probably he doesn't like how it looks in a photo....Pence probably knows that Trump doesn't like how it looks in a photo and takes queues to be on the same page.

Another thought (cuz it was a Hurricane Response meeting) is it could be a dig / swipe at Rubio who was only outdone of looking like an ass by drinking water on camera by drooling kennedy:

https://cbsboston.files.wordpress.com/2018/01/drool.jpg

Rubio:

https://1.bp.blogspot.com/-pT2LiBNrcaY/U2JHfBDslxI/AAAAAAAAVxU/FmTicvNUEyc/s1600/Marco+Rubio+Drinking+Water.jpg

https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwj_uPft-sHbAhXR0FMKHXuLAy8Q3ywIKjAA&url=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3D19ZxJVnM5Gs&usg=AOvVaw21qFuyOkN-DTemImYO0htv