Tcl, 97 bytes

time {puts -nonewline [format %c [expr ([incr i]-1)%26+65]]
if int(sqrt($i))**2==$i puts\ ""} 676

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Tcl, 97 bytes

time {puts -nonewline [format %c [expr ([incr i]-1)%26+65]]
if int(sqrt($i))**2==$i puts\ ""} 676

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Tcl , 100 bytes

Requires Tcl >= 8.7 to allow use of lseq

lmap i [lseq 676] {puts -nonewline [format %c [expr $i%26+65]]
if int(sqrt($i+1))**2==$i+1 puts\ ""}

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Tcl, 98 bytes

time {puts -nonewline [format %c [expr ([incr i]-1)%26+65]]
if int(sqrt($i))**2==$i {puts ""}} 676

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Tcl, 97 bytes

time {puts -nonewline [format %c [expr ([incr i]-1)%26+65]]
if int(sqrt($i))**2==$i puts\ ""} 676

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Tcl, 101 bytes

incr j
time {puts -nonewline [format %c [expr ([incr i]-1)%26+65]]
if $j**2==$i {puts ""
incr j}} 676

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Tcl, 98 bytes

time {puts -nonewline [format %c [expr ([incr i]-1)%26+65]]
if int(sqrt($i))**2==$i {puts ""}} 676

Try it online!