# [BASH] , <s>580 349 308 303 291 286</s> 283 bytes
Found this very entertaining, thanks! First variant was 580 bytes, then 349, ..., now 283:

<!-- language-all: lang-bash -->

    s=$1 n=$2 N=$[n+n-1];f(){ b=$1;for((j=1;j<n;j++));{((c<N)) && T+=%s;[[ $4 ]]&&h=$4 S[$[i+$2/2]]=-;t=${s:c:1} S[i]=${t:-$b} b=$3;((i+=$2,c++));S[$[i-($2/N)]]=\ ;};};while((n>0));do f . 2 - \>;((n-=a));f . $N \|;f \' -2 - \<;((n--));f \' -$N \|;a=1;done;S[i]=$h;printf "$T\n" "${S[@]}"

[BASH]: https://www.gnu.org/software/bash/
[Try it online!](https://tio.run/##JZBNboQwDIWvYqGUJkrTFqZSJUJGPQGbmV3Igl8ROjIVMOqCcnZqGGVh5z1/dpyymLptmwyLAA2LITPMokQVOd1ysUBJjm6HkfPeRLpPUfdSCqEXzqs0EwLCEK7SPE3aWmAf4FwYdoaSi2XWSxa/xc4ZpWfDlimpkmglxzu6zYli5boPOGnOvaTpL9XR@0AVJzYTBOegVzq/nb81nOP5nUrqAVp4hRgU5GfCUZmC5F1jGeR/lOXPoA4/PXx12Lv4KChonXrARj@e0@mf0ePcQsCuOQYUlov9cmuw0e9g8d0U43DHur1jNfsBCbx5bMbt8x8)

Squeezed to 283:

    #!/bin/bash
    s=$1 n=$2 N=$[n+n-1]
    
    f(){
        b=$1
        for ((j=1; j<n; j++)); {
            ((c<N)) && T+="%s"
            [[ $4 ]] && h=$4 S[$[i+$2/2]]=-
            t=${s:c:1} S[i]=${t:-$b} b=$3
            ((i+=$2,c++))
            S[$[i-($2/N)]]=' '
        }
    }

    while ((n>0)); do
        f .  2    - \>; ((n-=a))
        f .  $N  \|
        f \' -2   - \<; ((n--))
        f \' -$N \|   ;   a=1
    done; S[i]=$h; printf "$T\n" "${S[@]}"

Reduced to 349 bytes:

    s=$1 n=$2 N=$[n+n-1]; for ((p=0; p<n*N; p++)); { S[p]=" "; }; printf -v T "%${N}s"; T=${T// /%s}; f(){ b=$1; for ((j=1; j<n; j++)); { [[ $4 ]] && h=$4 S[$[i+$2/2]]=-; t=${s:c:1} S[i]=${t:-$b} b=$3; ((i+=$2,c++)); }; }; while ((n>0)); do f . 2 - \>; ((n-=a)); f . $N \|; f \' -2 - \<; ((n--)); f \' -$N \|; a=1; done; S[i]=$h; printf "$T\n" "${S[@]}"

[Try it online!](https://tio.run/##PVDLboMwEPyVEXLSpNQloZUqxXHUL@ASbsYHkoBwGi0RkPZA@Xa6zqOSZY9nZsfr3eVtNY6tFkuQFjESLQyFJJdWoawbzGZnvVA4r@k54SMM53OFHltztjpAoDAw2zjqSshvpAgmok@GloVUiz6NIkSTlj3lbN5jx888Yo@a4XFNvD1CjYF4h7WYTlFphlsjjAtFHMXWaqnQcWS72q@WA0vO8q1bSbEbfPCb4lAX8h9e9rfA4bp@KncqWKLNwpOHGiVeEUMi2/gSkjr3gmdFguzXw@wJ8mpZ3yzy5vD03ZP79g81FereSvU/h0CkGfFsRL81n3YIRp4v5V9F3tQXOpQX2neuJi49OSqa8eMP)

And ungolfed for better readability:

    #!/bin/bash
    s=$1 n=$2 N=$[n+n-1]
    for ((p=0; p<n*N; p++)); { S[p]=" "; }
    printf -v T "%${N}s"; T=${T// /%s}

    f(){
        b=$1
        for ((j=1; j<n; j++)); {
            [[ $4 ]] && h=$4 S[$[i+$2/2]]=-
            t=${s:c:1} S[i]=${t:-$b} b=$3
            ((i+=$2,c++))
        }
    }
    
    while ((n>0)); do
        f .  2    - \>; ((n-=a))
        f .  $N  \|
        f \' -2   - \<; ((n--))
        f \' -$N \|   ;   a=1
    done; S[i]=$h; printf "$T\n" "${S[@]}"

First variant was:

    #!/bin/bash
    
    s=$1
    n=$2
    l=${#s}
    N=$((n+n-1)) # items per row
    
    # Check if n is in range
    (( (n*n)>=(l+1) )) || { echo "n must be 'n^2≥l+1'"; exit 1; }
    
    # Fill the 'area' with spaces
    for ((i=0; i<n*N; i++)); { snake[$i]=" "; }
    
    # Create printing template
    printf -v row "%${N}s"
    row=${row// /%s}
    
    # Setcounters
    i=0
    c=0
    a=0
    
    
    forward(){
        b='.'
        for ((j=1; j<n; j++)); do
            cut=${s:c:1}
            snake[$i]=${cut:-$b}
            snake[$((i+1))]="-"
            b='-'
            h='>'
            ((i+=2))
            ((c++))
        done
        ((n-=a))
    }
    
    
    down(){
        b='.'
        for ((j=1; j<n; j++)); do
            cut=${s:c:1}
            snake[$i]=${cut:-$b}
            b='|'
            ((i+=N))
            ((c++))
        done
    }
    
    
    back(){
        b="'"
        for ((j=1; j<n; j++)); do
            cut=${s:c:1}
            snake[$i]=${cut:-$b}
            snake[$((i-1))]="-"
            h='<'
            b='-'
            ((i-=2))
            ((c++))
        done
        ((n--))
    }
    
    
    up(){
        b="'"
        for ((j=1; j<n; j++)); do
            cut=${s:c:1}
            snake[$i]=${cut:-$b}
            b='|'
            ((i-=N))
            ((c++))
        done
        a=1
    }
    
    
    print(){
        printf "$row\n" "${snake[@]}"
    #    echo $i,$c,$n
    }
    
    
    # Create snake
    while ((n>0)); do
        forward
        down
        back
        up
    done
    snake[$i]=$h; printf "$row\n" "${snake[@]}"

[Try it online!][1]


  [1]: https://tio.run/##vVTBjtowEL37K6bGu0mWTbPhUgnwqtJKPXLpsd1KjjEbL2ESxaGpBHxAP6Q/1h9JJ4aFtqirnrCU4BmPZ9570SNTLu@6wZsks5hkFDDmpEgZSjFihRSbgduxmRRhiEOM0yiCAdjGrBxUpoa6bBkbwENu9BLsAhCsA4tQK3wyLAwhxBuM7mVYDNMI6PJ2CxswOi@BI6zWroHMQIBfRj@//6CagE/AfLMNpBPY9Z0/2KKAJqcaVRsVQGubHFyltHFsUdYQhlbeTcBO8WZGP8NhFE1ogkO1NJ@EfZQc@KHVAzVoDFS1xcbiExCJqqAM85kFxF97OsCvxGa2c5xRQPTpnSSQXDnf46NpdLnGxtSO0WCm6VH0sB5Mq@p5GG0Y0Mpk8Dbwuz3KZ0mMnqdIrz3GeelP@6XXDQ1yYz1Od8fkiYHYUME4FtnZIZEnWSMiGfPjGU2Og2OUy@D@FPUX5CiKfkvoHo6P5yUatk9iLBVliTKbly1ehhR13/4FdfYa1B5dpvTyhI4H/AKSx2eSk8jT4B8foL/wf5LHB8XX1WUYnekdv653v1Ey9Ri9ZV5gHvzDBXnlM5LhCIcf/P5xx8k0tLzjhb0V@lag73D0oy9lbW4LA@MjpYOdDtNb3OtBX9tv1tWLalN5R/8q19d/eF7kE8io@7J3vsfuIXZd52tUTRae63Jl3H7bvfsF