The EPS light came on my 2016 Hyundai elantra and it won't turn on !!!! Why ??

There is a video here

https://www.google.com/search?q=replace+a+tail+light+in+a+2016+Hyundai+Elantra

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https://owners.hyundaiusa.com/content/dam/hyundai/us/myhyundai/glovebox-manual/2016/elantra/2016_Hyundai_Elantra_OM.pdf
pages 7-81 7-82

The electronic power steering (EPS) Warning Light is located on the instrument panel. This light illuminates when the EPS is malfunctioning. When this light is lit, your steering may be a little heavier, and you should schedule an inspection as soon as possible. Tip The EPS Warning Light briefly illuminates when the car is first turned on. The light is only a problem if it stays lit for an extended period.

The electronic power steering (EPS) Warning Light is located on the instrument panel. This light illuminates when the EPS is malfunctioning. When this light is lit, your steering may be a little heavier, and you should schedule an inspection as soon as possible.

Visit the on-line store, known as reliable-store, they have the repair manuals. I'm sure you will get the solution of your problem.
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EPS- Electronic power steering warning light. is there any steering problems ? Light being on means there probably is a trouble code in the EPS module .

why EPS lights still on while driving and did this problem during driving?

The "EPS" stands for Electronic Power Steering. There are a couple Technical Service Bulletins related to the EPS system published by Hyundai. The only way to find out if any of these bullitins apply to your problem, you must FIRST scan your onboard computer network to gather the fault codes that set when the system turned the light on. Then again, there may also be a problem that is not addressed by ANY of the bulletins.....Proper diagnosis required.

The same thing applies for your Check Engine light. There are currently over 600 possible reasons why the check engine light could be coming on in that vehicle. Many of these possible faults can cause your engine and transmission to go into "reduced power" or "limp-in" mode and cause the symptoms you are describing. The ONLY way you are going to find out which possible reason is causing the light to come on is with a computer network scan....Unless you really want to work your way through over 600 "GUESSES".

Sometimes (few) a fault in the wiring loop causing it to short on the steering column and turn on EPS (Electronic Power Steering), check it and scanned for know if some additional fault code is stored there.

Keep us updated.

Your Electronic Power steering has an issue - have a shop look at it for you before it fails completely.

Hello... This light indicates that you might have a problem with your Electronic Power Steering (EPS)...

Have a shop check it out...