B757-200
Flight Crew Air Transport Pilot (ATP)
Flight Crew Last 90 Days 120; Flight Crew Total 18500; Flight Crew Type 9800
Aircraft Equipment Problem Critical
Flight Crew Landed in Emergency Condition; Flight Crew Landed As Precaution; Flight Crew Diverted; General Declared Emergency
Aircraft
This flight began with an aborted start of the LH engine. The EGT exceeded the start limit to 523C. The 'Aborted Start' checklist was accomplished. We were towed back into the gate and a call was placed to Dispatch/[Maintenance]. An AMT arrived at the aircraft and after some discussion with [Maintenance]; it was decided that the exceedance was within limits for continued operations. The subsequent engine starts were normal and we proceeded for takeoff. Shortly after becoming airborne; the RH engine EGT advanced into the red and peaked at 650C before we retarded the throttle. The First Officer was flying and continued to follow the departure as I located and performed the 'Engine Limit; Surge; Stall' checklist.  While accomplishing this checklist; the EGT again exceeded red line limits.  Several attempts were then made to use the 'Call Me' prompt in ACARS.  Each time an error message was received. The message indicated that this function was not available. A text message was then sent to Dispatch requesting them to call.  Radio contact with our Dispatcher was accomplished through ARINC. We asked that [Maintenance] be added to the conversation. Unfortunately; [Maintenance] could not hear or understand our transmissions. They asked that we repeat each transmission several times. This became quite a distraction/threat and after several frustrating minutes we asked that they be dropped from the call. We continued the analysis of our situation with only Dispatch on the line. We concluded; that we did in fact have a degraded engine and the appropriate course of action was to divert to [a nearby airport].   We informed ATC that we would be diverting and an emergency was declared.  At this point; the Flight Attendants were brought up to speed about the condition of our aircraft and that we would be landing. This was followed by an announcement to our passengers explaining where and why we were going to land.  We then learned from our Purser that we had a [company] 757 crew in the cabin and they were willing to come to the cockpit if we needed assistance. We decided that an extra set of eyes and ears could be helpful. The deadheading First Officer came to the cockpit and backed us up with our checklists and subsequent approach and landing.  While this transpired; my First officer prepared the aircraft for arrival.  We were then advised that due to VIP operation; holding was in progress. We re-emphasized the emergency status of our aircraft and were given approach priority. While descending; another non-essential distraction occurred when we received several messages from [Maintenance] asking for specifics about our engine.  We responded later on the ground.  Since the overtemp only occurred at higher power settings; a normal two engine approach and landing was made.
As I was flying the aircraft and communicating with ATC; I was not completely aware of the issues the Captain was having with communication but it was distracting to both of us and it appeared as if [Maintenance] could not hear any of the Captain's transmissions.
B757-200 Captain reported high EGT in the engines both at start and shortly after takeoff.  An emergency was declared and the flight diverted to a nearby airport.