For example, "我不能睁开我的眼睛" and "我睁不开我的眼睛" (不 obviously being the negation of the 得 potential form). What is the difference in nuance between these two sentences?
This sentence means, "I cannot open my eyes," with the implication that the speaker is not allowed to or otherwise constrained.
This sentence means, "I am physically incapable of opening my eyes." (睁不开 sounds like, "[my eyes] are unopenable").
The 得／不 potential form is used with result verbs. As in your example:
我睁不开眼睛 ---> 开 (open, separated) is the result of 睁
Therefore the potential form tends to express an objective ability or inability to do something. Eg:
我打不开门 I can't open the door (it is too heavy, blocked, broken...)
我睡不着 I can't sleep (too anxious, there's noise...)
今晚我做不了饭 Tonight I can't make dinner (I'm too tired)
The 能 form sounds more like a subjective ability. It is used with generic actions that don't have a result (in the sense of a result complement), polite requests, to express moral/legal constraints. Eg:
你能想象吗 Can you imagine it? (subjective non-physical ability)
你能帮我忙吗 Can you give me a hand? (request)
今晚我不能做饭 Tonight I can't make dinner (generic action)
我丢了驾照，我不能开车 I lost my driver's license, I can't drive.
The opposite of '我睁不开我的眼睛'(I can't open my eyes) is '我睁得开我的眼睛'(I can open my eyes)
[不] and [得] are [potential particles]. They have to be placed after the verb.
For example: 去得到/去不到
去(go) is the verb; 得/不(can/cannot) is the potential particle; 到(arrive) is the resultative particle
The opposite of "我不能睁开我的眼睛" (I can't open my eyes) is '我能睁开我的眼睛'(I can open my eyes)
[能] and [不能] are [auxiliary verbs] (can; able to/ can't; unable to). They have to be placed before the main verb
For example: 能去 / 不能去
能(able to/can) is the auxiliary verb; 去(go) is the main verb
我睁不开我的眼睛 -- I'm not able to open my eyes
我不能睁开我的眼睛 -- I cannot open my eyes (even though I am able to do it. Maybe I do not want to see something, or be threatened not to do so, etc.)
The first sentence may have two possible meanings: that one is not able to open his eyes, or that he is not allowed to. The second one may only infer the first meaning noted above.