I cannot get reception on the flexible antenna which came with the Casio JY-70B Portable Color TV. I need the users manual

Refer its user manual to get more details about whether this TV is compatible to both analogue and digital signal reception. If yes, you needn't have a converter. If no; you should have. Go to the specification page of the user manual to get this detail.
http://electronicshelponline.blogspot.com/

radio shack has a adapter part # 278267 which converts the antenna coax cable to 1/8 or 3.5mm jack for the external antenna input my casio tv-970 works with a digital converter box

Get a digital converter box. Ha! Ha!

You won't. Because the unit isn't likely to have an coaxial external cable input, there is no simple way to achieve this. Full-size sets not equipped for digital signals can accept the down-converted signal from a converter box that receives the digital signal and converts it to NTSC standards for older sets. Direct reception of the new signal requires an ATSC tuner and there is no way to provide that for your hand-held. It would be possible to use the output of a converter box clipped onto the antenna of your Casio and see the channels the converter box can receive but your hand-held wouldn't be very portable anymore. Sorry-

http://www.funai-corp.com/support/manuals.aspx
Click on link to download the manual for your tv.
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Btw, I’m available to help over the phone in case u need at https://www.6ya.com/expert/jason_77a19afa8d604dee

You may have to use foil or other metal to extend the antenna artificially. I found years ago that even wrapping my hand around the end of it boosted the signal (though I'm sure you're not willing to listen to a radio while holding the antenna the whole time). I once sat a TV antenna on top of a portable CRT with its own internal ferrite antenna, and the video signal was about 80% there, audio 100%. This may be an inconvenient method, but it will guarantee at least most of the audio quality you want.

an external or outside antenna will improve the amount of stations picked up and maybe a amplified antenna will help
but the boxes they sell seem to do better in reception even with only a rabbit ear antenna.
I use a box on my 32in LCD because the internal tuner just does not get all the channels without having to get up and move the antenna and the outside antenna , well I'm not going to go out there to move it all the time , and the amplified antenna I still have to keep getting up to move the antenna
so in all the box( "zenith","digital stream"are good , have yet to test others) works the best in recieving the digital channels without much worry of the antenna

The converter box needs an antenna, but not the one built into the TV. In fact, you should get an antenna designed for the new DTV broadcasts to get the best reception. There are a number of styles and they start at under $10.

To connect the converter box to the TV, you have a couple of options. You can use the channel 3 or 4 output from the box and connect the cable to the TV external antenna jack with the appropriate adapter. Or, since this TV also works as a monitor, you could use the converter box video and audio outputs instead. If the TV has a single A/V input jack instead of separate audio and video jacks (the usual case), Radio Shack and other vendors sell the adapter cable you'll need. One might have been supplied with the TV.

I have my handheld Casio TV-980 up and running with a converter box. All I needed (in addition to the converter box and standard rabbit ears [UHF/VHF] antenna) was an adapter with an F jack on one end and a 1/8" plug on the other (Radio Shack catalog # 278-257, about $5). The 1/8" plug on the adapter goes in the external antenna jack on the TV, and the other end connects to the coaxial cable that connects to the TV (RF out) jack on the converter box. The adapter would have been a little cheaper at my local, independent electronics store, but they were sold out.
 
If your antenna has flat 300 ohm wire connections, you'll need a balun (Radio Shack catalog # 15-1297, about $8) to connect it to the coaxial cable that goes to the Antenna (RF in) jack on the converter box.
 
I’ve tried 2 different converter boxes with the handheld TV, and they both work well: Insignia (Best Buy’s house brand, same as Zenith) and RCA (from Target, same as Venturer), so any government coupon-eligible converter box should work.
 
A few other tips:
 
1. I always turn on the converter box before I turn on the TV. The cpnverter box seems to take control better that way.
 
2. When I turn on the TV, the tuner doesn’t automatically go to channel 3 or 4 (where it should be set when using a converter box), and the picture quality is poor. If I use the tuner on the TV to get to channel 3 or 4, the picture quality is vastly improved. I have to do that every time I turn on the TV.
 
3. Placement of the antenna is very important. I like to watch my handheld TV when I'm on the computer, and the reception is better the further away from the computer the antenna is located. I'm actually using a 6' coaxial cable for the antenna, but I could also have used 2 shorter cables with a coupler that has an F jack at both ends (Radio Shack catalog # 278-304, about $3).
 
If your battery-operated TV does not have an external antenna jack, instructions for connecting it to a converter box can be found at
 
http://www.digitaltvtrainer.com/national/showme-battery-tv-converter
 
There are a few battery-operated converter boxes available, but for do-it-yourselfers, this web page also contains instructions for building a 12-volt converter box power supply.

Check if u are not on A/V mode, return to RF/TV mode and try tuning again, also check your antenna connexion.