Yes. A cleric, druid, or paladin can choose to follow a cause they passionately believe in as opposed to a deity.
The PHB (see Appendix B) notes that it is perfectly acceptable for a DM to decide that there are no gods that are worshipped in his or her campaign:
Your DM determines which god, if any, are worshipped in his or her campaign.
Emphasis mine. In these instances where the players inhabit a faithless world, the clerics, druids, and paladins don't just disappear because there are no imaginary sky people to give them super powers. They draw their power from a devotion to whatever ideal they value above all others.
The DMG goes on to codify this a little more explicitly, and provides guidance on alternative religious systems to the standard polytheistic pantheon that the vanilla game assumes as the default. It even specifically discusses "forces and philosophies" as an alternative to deities entirely on page 13. It has a decent amount of text, so instead of quoting it in its entirety, I'll simply leave it to you to look up at your leisure. You might also consider animism, which the DMG discusses on page 12. Perhaps that storm was caused by an Air Spirit, and the act of walking through the storm without praying to it for safe passage angered the spirit, and now it calls the character to its service. There are many possibilities that are well-supported by the rules and do not require any substantial effort on your part to implement.
All clerics can use healing magic.
Per the descriptions in the Player's Handbook, a cleric's magic comes from their deity:
Divine magic, as the name suggests, is the power of
the gods, flowing from them into the world. Clerics are conduits for that power, manifesting it as miraculous effects.
(PHB, p. 56)
Clerics can serve an evil god. Whether a player character cleric can serve a particular evil god is up to the DM to decide; some evil gods may not require all of the their servants to be evil, or to do evil deeds.
All clerics, regardless of the alignment of their deity, use the same basic spell list, which includes a number of healing spells. Each cleric chooses a divine domain which is related to their deity; this will give them additional spells and abilities. The DMG provides rules for an additional domain (Death, on p.96), which may be appropriate for clerics of some evil deities.
Note that there's no reason that a cleric of an evil god shouldn't be able to use healing spells. The cleric, their allies, minions, and followers, can all benefit from healing, and healing their servants may be entirely in the interests of an evil god.
Best Answer
Ask your DM:
That is all you or I know about this. We can't tell you what form it will or can take, because the literal rule is that the DM gets to make something up that seems appropriate to the exact situation and your specific request at the time. Whether or not the intervention of an evil god would be different from the intervention of a non-evil god is included in that βit's up to the DM.β
So, you'll have to ask your DM β and be prepared for the possibility that your DM will respond that you'll have to try it during play to find out.