with one's eyes shinning と with one's shinning eyes (with shining eyes) ではニュアンスが違います。
with one's eyes shinning が「目を輝かせながら」「目を輝かせて」なのに対して、with one's shinning eyes (with shining eyes) は「輝く目で(の)」といった感じ。( with (one's) green eyes が「緑色の目で(の)」というのと同じ様な感じ )
※また、文脈上誰の目か分かりにくいとか、「誰の」を強調する場合以外は one's の無い with shining eyes のほうが普通なようです。
1) with shining eyes
Sharing our rickety table, under a canopy of stars, Metin told us with shining eyes of his love for the island, which in winter he and five or six other families have almost to themselves.
http://travel.independent.co.uk/europe/article768.ece
The next day a tall teenager with shining eyes came to visit.
http://www.bbc.co.uk/dna/getwriting/A2900206
When they reached the big plate-glass window of the big cats enclosure, one little girl looked, gasped, and turned to her teacher with shining eyes and said, "But Miss, you never said the tigers would be REAL."
http://books.guardian.co.uk/news/articles/0,6109,1543346,00.html
2) with one's shinning eyes
When I think back I can still see the eager young friends I made and lost, with their shining eyes and such hope for the future.
http://www.bbc.co.uk/ww2peopleswar/stories/77/a3889777.shtml
In contrast Mr Blair, with his shining eyes and happy-clappy manner, has always looked pretty batty, albeit in a reasonably harmless way.
http://www.guardian.co.uk/blairbabe/article/0,,184546,00.html