LTC2400 has DC573A and DC590B to get a quick start, but LTC24001 doesn't have such kits.
Can I transfer the data collected by my designed LTC2401 board to the PC for analysis and storage by using a kit like the DC590B?
The LTC2401/LTC2402 are 1- and 2-channel 2.7V to 5.5V micropower 24-bit analog-to-digital converters with an integrated oscillator, 4ppm INL and 0.6ppm...
LTC2401 on Analog.com
You should be able to use the LTC24XX Linduino Code with the DC2026 to capture data and then store it in the PC.
To use your board and the DC590 with QuikEval you would have to emulate the eeprom of a similar LTC24XX demoboard with QuikEval support. I'm not sure how easy that would be.