Non-religious equivalent expression for "Pray for [Country X]" after a disaster
Solution 1:
Since praying, in this context, is a religious activity, there is no non-religious equivalent.
However you might use "Spare a thought for [X] [at this difficult time]"
Solution 2:
Pray does not only have a religious connotation: (from M-W)
to hope or wish very much for something to happen
to seriously ask (someone) to do something
but if you prefer a different expression; give your emotional support to (Country name), may fit the context.
Solution 3:
I wish them well myself.
It's a lot like praying for them, but less diety oriented.
Solution 4:
A non-religious equivalent to "Pray for Country X" would be "Let our thoughts be with Country X."
Solution 5:
Yes, among Christians at least, the phrase "pray for (someone)" is literally a request that you say a prayer for their health/safety/recovery/etc.
There isn't a literal equivalent that avoids including prayer, but a similar secular phrase would be to "keep (someone) in your thoughts".
These often get combined into: "keep (someone) in your thoughts and prayers" to include (or at least hopefully avoid offending) both those who pray and those who do not.