Ken,
Make sure you change BOTH of the "Y=Y+1" statements to "Y=Y+2"

I have verified your code works if you change both. Here is my version of your code. I've trimmed it down, modified the output methods, and generated the temps without the ADC. But your will notice all the WRITE and READ statements are the same. I output the values in hex, which allowed be to then read the EEPROM as an additional way to verify the data.

Code:
TEMPF   VAR WORD
TEMPC   VAR WORD
Y       VAR BYTE
X       VAR BYTE
Z       VAR BYTE
TEMPF1  VAR WORD
TEMPF2  VAR WORD


START:
    PAUSE  2000     
    Y = 0  
    for x = 1 to 5
 
WRITE_temp:      'READ THE THERMOCOUPLE AND SAVE IT
       TempC = 1 << Y
       TempF = (TempC*18)/10+32
       LCDOUT $FE, $80, HEX4 TempF
       pause   30    
    '      READ THIS TEMPERATURE
       HSEROUT ["TEMP= ",HEX4 TEMPF, 13]
'
'       NOW WRITE IT TO EEPROM LOCATION IN VARIABLE Y
       WRITE Y,WORD TEMPF
'
'   IMMEDIATELY READ IT BACK  - - THIS VALUE IS READ CORRECTLY ALL 5 TIMES
'
       READ Y, WORD TEMPF1 
       HSEROUT ["READ TEMP= ",HEX4 TEMPF, 13]       
'******************************    
'    NOW READ IT BACK AND WRITE TO A DIFFERENT VARIABLE ( TEMPF1)
'   THIS READ ALSO WORKS CORRECTLY ALL 5 TIMES
'*****************************  
       READ Y, WORD TEMPF1
       HSEROUT ["READ1 TEMPF1= ",HEX4 TEMPF1, 13]
       
       LCDOUT $FE, $C0,"X = ",dec x
       LCDOUT " Y = ",dec Y
      
       Y = Y + 2
       next x       
       
       PAUSE   3000
       HSEROUT ["-----------------------------", 13]

'**********************************
' NOW TRY TO READ IT AGAIN, USING ANY VARIABLE DOES NOT MATTER , TEMPF1, ETC
'    THE ONLY CORRECT READ HERE IS THE LAST ONE, (Y = 4) 
'ALL OTHERS ARE INCORRECT - - VERY HIGH VALUES - - 23375, 22860, ETC
'*****************************************************************
       Y = 0         'RESET THE EEPROM ADDRESS POINTER TO 0
       FOR X = 1 TO 5 
       READ Y, WORD TEMPF2  ' USING VARIABLE TEMPF2 HERE, ANY OTHER VARIABLES 
                            ' DO NOT WORK EITHER
       HSEROUT ["READ2 TEMPF2= ",HEX4 TEMPF2,13]
       Y = Y + 2
       next x      
             
       HSEROUT ["Done", 13]
DONE:
     
      pause   1000    
      GOTO   DONE

    
END