I checked the New Brunswick Land Petitions 1774-1997. There is only 1 Eager listed-Windsor Eager in Cumberland Township,County Sunbury. Date granted is 1786-10-05. Right date for a Loyalist. The province charges $8.00 for a copy.
It might be of interest to you that there are 21 Eager surnames listed for Lower Canada Land Petitions & 6 listings for Upper Canada Land Petitions. All these Land Petitions are digitized on line from Archives Canada & the best part is they are FREE. Just Google Lower Canada Land Petitions or Upper Canada Land Petitions & follow the links. Glad to help further if needed.