What size supercapacitor do i need to replace 4 3.7v 2400mah batteries

Charging NIMH (and most modern battery chemistries) isn't done at constant voltage, but at constant current.

You'll need some fairly involved electronics to provide a constant current source and monitor the battery state.

Battery University has a pretty good description of the charging cycle of NIMH batteries.

The initial charge is at 1C (in your case, 175 mA). There are 3 different stop-checks (whichever is reached first):

1. Temperature rise rate of the cell being charged.
2. Voltage across the cell.
3. NDV (negative delta voltage). Basically check for when the voltage rise of the battery stops rising at a certain rate.

At this point, the charger will charge the battery at significantly less current, repeating similar 3 checks and iteratively stepping down the current charge rate.

The linked page suggests that slow-charging a NIMH battery is difficult/impossible because it's very difficult to get reasonable readings for many of the time-dependent stop checks. You're basically relying on a time charge, which could excessively charge the NIMH cell and damage the cell.

The primary safety concern is excessively heating up the battery. You might risk rupturing the battery, though I don't think NIMH are susceptible to catching fire as some other cell chemistries are. Most likely the battery casing is designed to rupture in a safe manner, but no guarantees.

It's unlikely you'll do anything which would be considered a significant safety concern, just use your common sense. There aren't any high voltages involved, and the power outputs are relatively low.

Under charge currents of << C/1 a Nimh cell is fully charged at about 1.45V. Using 1.4V/cell gives you slight lee way at the loss of a small amount of capacity. Slightly lower again is even safer.

If you don't mind wasting solar energy (and that should not be a problem here), feed the batteries through a diode (if there is not one in the panel already, and clamp the battery voltage at 5.6V (1.4V/cell). [ Or 5.4V at 1.35V/cell for good safety].
I assume battery temperatures are in the 20 -30C range usually - best voltage will vary somewhat with temperature but that should work well enough.

That panel is perhaps 2 Watts (you may have a spec there or can measure short circuit current in full sun). If so then Imax is about atts/Vmax_power = say 2W / 15V =~~ 130 mA. Actual could be 50 200 MA - neither extreme being too likely.

100 - 200 mA is too much for a cheap TL431 clamp regulator by itself.
A TL431 driving a TO220 P Channel MOSFET (plus a few resistors) or an N Channel MOSFET plus any small PNP transistor will give you a clamp regulator suitable for the task.

TL431 divider string top resistor can be maybe 100k so drain on battery when there is no sun is around 50 uA = not a problem in this application.

Or you could use a standard series regulator such as an LM317 plus 2 resistors set to 5.6V would work but backfeed via the regulator and resistors adds slight complexity.

There are other ways but the TL431 + MOSFET clamp should work well enough.

TL431 "turns on" when gate voltage >= 2.5V.
Z1 on pulls Q1 gate low turns Q1 on which dissipates excess energy in Q1 + Rload.
Rload is optional if MOSFET can dissipate all energy OK - but usually using a resistor avoids needing a heatsink.
Rload = V/I <= (Vbattery_max - V_FET_on) / I_panel_max
V_FET_ON is the voltage drop across the fully on MOSFET
= Rdson x I_panel_max.
With a MOSFET with Rdson = say 0.1 Ohm then on voltage at say 150 mA = = V = IR = 0.15 x 0.1 = 15 millivolts, so a half decent FET needs minimal allowance for on voltage.

Say V_FET_on = 0.1V, Imax = 150 mA, Vbat max = 5.5V.
Rload = V/I = (5.5 - 0.1) / 0.150 = <= 36 Ohms. 33 Ohms OK.
Lower OK but FET will then make up some of load and dissipation may be higher.

^{simulate this circuit – Schematic created using CircuitLab}

LM336 datasheet - All LM336 datasheets I found were poor (Faichild, TI, LT).NONE gave adj pin current.
If Vclamp is not correct R3 & R2 may need to be lowered while maintaining 1.2:1 ratio.
eg 33k : 27K