"on or after the Participant's attainment of the earliest retirement age"
Everything is governed by the plan. Ask for a copy of the Summary Plan Document (SPD), it should clearly state the plans earliest retirement age. If that is age 62 or there is no Early retirement date, than you are not eligible to start your benefit before the participant's age 62. As far as benefits express, they are in no way subject matter experts. They may not even have access to a full plan document and handle several clients and plans at at once. They can be easily confused and give out incorrect infomation.
Yes, you can send an official inquiry "claim for benefits" and it will have to be reviewed by the company. However, they won't make an exception if the rules clearly state the early retirement date is 62, even if you were initially given the wrong information..