[kwillisjr]

Do I need to ground the phono input to the screw on the board or can I use one of the RCA grounds on the opposite side of the board?

[poty]

I'd follow JH initial design: it is well thought and professionally implemented.
IMHO, the RCAs are far from the best place of grounding.

[hagtech]

You can connect the turntable tonearm ground to the screw. 

jh

[kwillisjr]

Since I will be using separate metal enclosures for the power supply and bugle, do I need to earth ground the bugle enclosure?  If so, do I need to keep the board ground isolated from the chassis (earth) ground?  Or, in other words, do I keep the RCA and turntable grounds isolated from an earth grounded chassis?

[hagtech]

It is usually best to ground the chassis to earth, and then to board at a single spot, usually at or near the inputs.

jh

[poty]

One way or another - you should use the opportunity to shield the Bugle (which have very sensitive circuits).
What to say about connecting the metal enclosures to the ground point on the PCB. I depends on what you chose as your ground schema.
"Classical way" - you ground everything to the mains gound. In your two-enclosure design you connect the mains ground near the AC entry to the enclosure, then connect the dedicated wire to the ground lug in the second enclosure (non-isolated). The ground point from the PCB then you connect to the same ground lug. RCAs should be isolated from enclosure and connect to the ground lug only through the PCB.
You may ground "signal circuits" through the interconnects to the next device - this way the grounding works when you powering Bugle from batteries. In the case you go this way you must connect the mains ground to the enclosures only, ground lug should be isolated as all RCAs too.
There are many other ways to make grounding, in a message on the forum here someone mentioned a document explaining theoretical and practical approaches to the design.