Being longsighted will mean a childs eyes are prone to have a convergent (eye turns in) squint develop if tired.
The specs normally will keep the eyes straight if it is the type of squint which is called an accommodative squint which is most common if the child is very longsighted.
She will see without spectacles as you say because she is young and therefore she has lots of accommodation (this is the ability of the muscle that holds the lens of the eye to change it's shape when focusing) but this is a real strain on the eyes if you are very longsighted as she is and she will be getting eyestrain/tired eyes and the eyes simply can't cope after a while hence why she 'breaks down' into a squint in one eye.
If she doesn't wear the specs there is a risk the squint could become more apparent even when she isn't tired and the eye lazy/weak permanently as the eye that turns won't be used with both eyes open.
If the specs alone control the squint she won't need patching.Patching is only if one of the eyes is becoming lazy which is usually because of a squint which is apparent or a big difference in the prescription of the two eyes .Having a big difference in prescription between the two eyes (anisometropia) could also cause one eye to become lazy but full time spectacle wear normally helps this along with partial patching sometimes.
The period for wearing specs all the time generally lasts up to about age 6ish which is called the plastic period in that this is the time thought to be when the vision is amenable to change and beyond this mostly if a eye is lazy it can't be fixed.
If you are very longsighted normally you will still need full time spectacle wear up to this period and beyond this accordingly sometimes reducing to partial wear as longsightness does improve as children get older as the eye grows along with them.
It sounds to me (I'm an optometrist) quite the right thing if she is very longsighted that she needs full time spectacle wear and this is really very critical if she is showing a squint when tired.Definately if this is the eye specialist's advice I would follow it.
I presume they tested her eyes using drops (cycloplegic) which will give her full prescription and they will check this again in 6 weeks.She would normally be assessed by the orthoptist (specialises in how the eye muscles work together) as well as the top bod the ophthalmologist (the eye medical specialist) and they will moniter carefully for any change in vision in the eye prone to developing a squint.If there is any sign of this they may then commence patching.
I wouldn't bother with a second opinion as this all sounds reasonable.
Well done to your Dh on picking up the squint.